A short characterization of the octonions

Abstract

In this article, we prove that if R is a proper alternative ring whose additive group has no 3-torsion and whose non-zero commutators are not zero-divisors, then R has no zero-divisors. It follows from a theorem of Bruck and Kleinfeld that if, in addition, the characteristic of R is not 2, then the central quotient of R is an octonion division algebra over some field. We include other characterizations of octonion division algebras and we also deal with the case where $(R,+)$ has 3-torsion.

Keywords:

• Alternative ring
• associator
• commutator
• octonion algebra

2020 Mathematics Subject Classification:

• Primary: 17D05
• Secondary: 17A35

Acknowledgement

We would like to thank Professor Holger Petersson for remarks that improved the results of this paper. We thank the referee for a thorough reading of the paper and for asking us to clarify certain points in the paper.